Question

What is the integral of \[\int {\dfrac{{\cos x}}{x}} dx\]?

Answer 1

Hint: To find the integral of \[\dfrac{{\cos x}}{x}\], we will use the Taylor series expansion of \[\cos x\] to expand. Then we will divide by \[x\] to find the expansion of \[\dfrac{{\cos x}}{x}\]. And finally, we will integrate this series term by term to give a power series expansion for the integral of \[\dfrac{{\cos x}}{x}\].

Complete step-by-step answer:
We have to find the integral of \[\dfrac{{\cos x}}{x}\].
We can’t do this integration in terms of elementary functions. We can do it in terms of infinite series using Taylor series expansion of \[\cos x\].
We know that the Taylor series expansion of \[\cos x\] is
\[\cos x = 1 - \dfrac{{{x^2}}}{{2!}} + \dfrac{{{x^4}}}{{4!}} - \dfrac{{{x^6}}}{{6!}} + ...\]
We require the integral of \[\dfrac{{\cos x}}{x}\]. So, we will divide the Taylor series expansion of \[\cos x\] by \[x\].
Dividing Taylor series expansion of \[\cos x\] by \[x\] gives us an infinite series expansion of \[\dfrac{{\cos x}}{x}\]. Therefore, we get the expansion as
\[ \Rightarrow \dfrac{{\cos x}}{x} = \dfrac{1}{x} - \dfrac{x}{{2!}} + \dfrac{{{x^3}}}{{4!}} - \dfrac{{{x^5}}}{{6!}} + ...\]
Now, we will integrate this series term by term which will give a power series expansion for the integral of \[\dfrac{{\cos x}}{x}\]. Therefore, we get
\[ \Rightarrow \int {\dfrac{{\cos x}}{x}} = \int {\dfrac{1}{x}} - \int {\dfrac{x}{{2!}}} + \int {\dfrac{{{x^3}}}{{4!}}} - \int {\dfrac{{{x^5}}}{{6!}}} + ...\]
Integrating right hand side of the above equation using the formula of integration, we get
\[ \Rightarrow \int {\dfrac{{\cos x}}{x}} = \ln x - \dfrac{{{x^2}}}{{2 \times 2!}} + \dfrac{{{x^4}}}{{4 \times 4!}} - \dfrac{{{x^6}}}{{6 \times 6!}} + ... + C\]
Where, \[C\] is the constant of integration.
Therefore, integral of \[\dfrac{{\cos x}}{x}\] is \[\ln x - \dfrac{{{x^2}}}{{2 \times 2!}} + \dfrac{{{x^4}}}{{4 \times 4!}} - \dfrac{{{x^6}}}{{6 \times 6!}} + ... + C\].
So, the correct answer is “\[\ln x - \dfrac{{{x^2}}}{{2 \times 2!}} + \dfrac{{{x^4}}}{{4 \times 4!}} - \dfrac{{{x^6}}}{{6 \times 6!}} + ... + C\]”.

Note: One important point to note here is that this is one of those integrals that can’t be done in terms of elementary functions. We can do it in terms of infinite series and we can use various numerical methods to do the definite integration. Also, this sum approaches zero so that the definite integral is \[\ln x\] up to an integration constant.